Probability maximizing approach to a secretary problem by random change-point of the distribution law of the observed process

Abstract

Before some random moment θ, independent identically distributed random variables x 1, · ··, xθ– 1 with common distribution function μ (dx) appear consecutively. After the moment θ, independent random variables xθ, xθ +1, · ·· have another common distribution function f (x)μ (dx). Our information about θ can be constructed only by successively observed values of the x's. In this paper we find an optimal stopping policy by which we can maximize the probability that the quantity associated with the stopping time is the largest of all θ + m – 1 quantities for a given integer m.